Tag Archives: Computer-Based Math

For Pi Day this year (2018), I will provide some results related to this interesting mathematical constant. These results are mostly inspired or taken from answers I gave at quora.com about or about similar mathematical topics.

The millionth decimal digit of π is1 (verified with Mathematica).

The 10 millionth decimal digit of π is found to be 7, and the 100 millionth decimal digit of π is 2.

The billionth decimal digit of (in base 10) is 9 (verified with Mathematica).

The 2 billionth decimal digit of π is found to be 0 (this result takes a longer time to compute with Mathematica).

Here are some (repeated) number sequences or numeric strings found among the first 2 billion decimal digits of π.

The featured image above was made with Mathematica . It represents the propagation of an electromagnetic wave.

Elements of this post were written for an answer I gave atquora.com , the question was about Maxwell’s equations
in tensor form. I have modified the answer and added material and equations related to General Relativity and gravitation.

Maxwell’s equationsare the fundamental equations of classical electromagnetism and electrodynamics. They can be stated in integral form , in differential form (a set of partial differential equations) , and in tensor form.

In the above equations , E is the electric field (vector field) , B is the magnetic field ( pseudovector field) , ρ is the charge density , and J is the current density.ε0is the Vacuum permittivity ,
and μ0is the Vacuum permeability .

The equations of electromagnetism (Maxwell’s equations and the Lorentz force) in covariant form (invariant under Lorentz transformations) can be deduced from thePrinciple of least action .

The electromagnetic wave equation can be derived from Maxwell’s equations , its solutions are electromagnetic (sinusoidal) waves.

All the steps and equations related to the topic of this post will not be shown here because they would make the post much too long , but an overview of the equations , calculations and derivations will be presented.

The development of the components of the Lorentz force and (after some calculations) its formulation in tensor form allows the introduction of the electromagnetic field tensor Fij:

where ujis the four-velocity four-vector .

The variation of the action δS with respect to the coordinates of a particle gives the equations of motion.

The action for a charged particle in an electromagnetic field can be expressed as:

q is the electric charge.

The variation of the action gives:

After some calculations and an integration by parts the variation of the action becomes:

In the equation above , uiis the four-velocity four-vector.

Since the trajectories of the particle are supposed to have the same initial and final coordinates , the first term in the right-hand side of the equality above is equal to zero. The potentials are a function of the coordinates , and the following equalities can be used:

Which gives for the variation of the action:

Thus the electromagnetic field tensor emerges from the variation above ; in fact the electromagnetic tensor is given by the definition:

Ai is the electromagnetic four-potential comprising the electric scalar potential V and the magnetic vector potential A .

The electromagnetic tensor is antisymmetric:

And the diagonal components in it are equal to zero:

The components of this tensor can be found using:

For example:

The others components can also be calculated and one gets for the covariant electromagnetic tensor:

I will continue with answers and results equal to the square root of 36 ( originally answered at quora.com ). This time the results are mostly related to physics.

With physics one has to take into account the units and the corresponding dimensions of the equations and of the constants.

6 and the square root of 36 are dimentionless numbers , so the result must be dimentionless . If the result is a simple fraction with numerator and denominator , then the units usually cancel out. In other cases when one deals with logarithms one should multiply with the inverse dimensions to get a dimentionless result. In one or two results where I didn’t look for the inverse units I multiplied the equation with a quantity I called (U) , which represents the inverse of the units by which one should multiply the result to get a dimentionless number.

Here are the results :

One possible way to explain what I have done here is the following:
If some people , living on an isolated fictitious island or on an another hypothetical planet , attached a great importance to and had a fixation on the square root of 36 (or the number 6) for one reason or another , and got accustomed to the use of 6 as a fundamental constant , unit or number , then they would have likely tried to construct a system of measurement based on the number 6 , and to express physics and math formulas ,equations , constants and rules in relation to 6.

After all , 6 oris equal to :

The floor of : ;

of the circumference of a circle in degrees.

It is also one tenth of 60 seconds which make up a minute , one tenth of 60 minutes which make up an hour , one fourth of 24 hours which equal a day on Earth , one half of 12 months which make up a year , etc.

A peculiar ‘hexacentric’ system , so to speak.

Or this can be seen as a (creative) exploration of or exercise in advanced math and physics in order to express many equations , formulas and constants in relation to the number 6(or) .Or whatever.

Does the future of humanity depend on answering what is the square root of 36 , or not?
Have philosophers from Antiquity to the present overlooked this fundamental question , which goes beyond the Kantian categories of space and time set out in his Critique of Pure Reason , and beyond Nietzsche’s Beyond Good and Evil , ushering the transmutation of all values and a defining moment for a new era in the history of Humankind?
It’s just a square root , for common sense’s sake (or is it?).

Anyway , enough philosophizing.

Here are () more answers to, this time with images :

is equal to :

— The number subjected to a geometric rotation in the following image (done with Mathematica and some Photoshop) :

— The number expressing the power and the coefficients in the equation of the curve in the polar plot below :

— The number expressing the degree of the root and the power of the variables in the 3D plot below :

The rotated number and the polar plotted curve in the first two images above seem to exhibit symmetry.
Symmetry is an very important property in science , math , physics , equations , nature , and wherever it is found.

I went over towww.quora.coma few weeks ago to answer a question about calendars , and then I got busy there. Since I am able to answer different types of questions , I started answering one question after the other , and I got stuck . I mean it’s a good way of getting stuck , answering questions about culture and science is useful and educational , but it can become time consuming and it requires attention and dedication.

Anyway . somebody came up with a question about the square root of 36. This question was obviously a stale unoriginal question , probably meant as a joke , but I decided to spice it up a little and make it more interesting.So I answered the question my own way , and I got a good amount of likes and ‘upvotes’ .

I will rewrite the answer I gave in here (with some modifications ).

Here are some results equal to:

An here is another group of results equal to .If one tries to work out or verify these equalities , it would be a good exercise in intermediate and advanced math ( and physics).

Given a date in the Islamic calendar , there is a formula which gives a good (approximate) numerical value of the corresponding date in the Gregorian calendar.

To find this formula or relation , we note that the ratio of a mean (lunar) year of The Islamic calendar to a solar year is:

The first year of the Islamic (Hijra or Hegira) calendar started 19th July 622 according to the Gregorian calendar. 19th July is the 200th day of the year and is (approximately ) 0.5476 in parts of the solar year ,while the number of years elapsed is equal to ( y-1) . The days are distributed regularly in both calendars , so the date of the beginning of the year y in Gregorian years is :

which can be written as :

(1)

Solving the equation above for the year of the Islamic calendar we get:

(2)

If we try to take a more accurate value for the average year length of the Gregorian calendar and take into account additional decimal digits , we get the following slightly more precise formulas replacing the two formulas (1) and (2) above :

yhjtoch(t) = 621.5773576247731 + 0.9702237766245703 t

and:

ychtohj(t) = -640.6536023959899 + 1.0306900573793625 t

Now if we solve the two equations (1) = (2) above or yhjtoch(t) = ychtohj(t) , we should find the date when the two calendars are equal and have the same day of the same month of the same year . Solving (1) = (2) gives the result 20875.052079 , which corresponds to 19 days (0.052079×365 ≈ 19) of the year 20875 , or 19th January 20875 , or 19/1/20875 .
yhjtoch(t) = ychtohj(t) gives the result 20874.956161756745132 , which corresponds approximately to the 349th day of the year 20874 , 16 days before the end of the year.
Using a calender converter (within Mathematica or an online converter ) , we find that the dates when the two calendars are equal lie between 1/5/20874 and 30/5/20874 , i.e the first day of the 5th month (May) of the year 20874 in the Gregorian calendar is also the first day of the 5th month of the year 20874 in the Islamic calendar , and this goes on until the 30th day of the 5th month of the year 20874 in both calendars. Therefore we can see that the dates obtained by making equal the two sets of equations above deviate and are further away from the real dates verified with calendar converters.

From the beginning of the Islamic calendar to the year 20874 there is a very small increase in the difference between the days of the year for the Islamic and the Gregorian calendar, and for the year 20874 the difference between the 2 calendars (if we equate ychtohj(t) and yhjtoch(t) ) is about 215 days , so I made some calculations and I subtracted a small term (0.000001758733 times t) from the formula converting from the Islamic to Gregorian calendar yhjtoch(t) and I got the following :

yhjch(t) is a little more accurate than the equation (1) above , which differs from the real date by approximately one day for years and dates in the current (21st) century , then the gap widens between yhjch(t) and (1) , with yhjch(t) giving dates a little before the real date , and equation (1) a little after the real (Gregorian calendar) date.For the year 20874 yhjch(t) gives more precise dates.

Setting ychtohj(t) = ychj(t) and solving ychj(t) = yhjch(t) , we get the result 20874.349006727632514 ,which is a date that lies within the interval between 1/5/20874 and 30/5/20874 for the two calendars.

Below is a graph showing ychj(t) and yhjch(t) around t = 20874 and how they intersect :

So the two conversion formulas or linear equations ychj(t) and yhjch(t) can be considered to be the two most accurate ones for converting between the Islamic and Gregorian calendars.

Additional reference work related to this post :Time measurement and calendar construction, by Broughton Richmond.

Results and things related to pi are usually published or made known on Pi Day . But you never know when you get inspired by π or find time to explore this ‘venerable’ math constant , and I have already published a post about pion Pi Day . By the way there is also a Pi approximation Day (July 22) , so there’s more than one date to talk or write about π .

Let’s kick off with an image of the value of pi (in the clouds ) with 12 decimal digits , made with Photoshop.I tried to make it realistic and show π and its numerical value ( 3. 141592653589) as part of the clouds and the sky.

Click on the image above to see and enlarged version.

Now for some computer-based mathematical explorations related to π.
There is a known relation between e ( the base of the natural logarithm ) and π :

and is approximately equal to 20 or almost 20 , which is known as an almost integer.
If we try to be more accurate and find the first 1000 decimal digits for the expression above we get (with the help of Mathematica) the following number:

19.99909997918947576726644298466904449606893684322510617247010181721652594440424378488893717172543215169380461828780546649733419980514325361299208647148136824787768176096730370916343136911881572947102843075505750157713461345968680161070464780150721176248631484786057786790083331108325695374657291368002032330492961850463283115054452239990730318010838062172626769958035434209665854687644987964315998803435936569779503997342833135008957566815879735578133492779192490846222394896357465468950148911891909347185826596341254678588264050033689529697396648300564585855142666534919457239163444586998081050100236576797224041127139639108211122123659510905094871070706680635934325684092946890616346767578519812785089761055789304041857980123101280905543416254404987679233496308302396952371198509012175432057419088516489412743155057902167919927734272964964116423666794634333328342687902907792168390827162859622042360176355034576875485783678406122447755263475337650755251536818489395213976127148481818560841182505647 ,
which shows that does not approach 20 completely or uniformly.

Here are some more expressions and calculations involving π (calculated with Mathematica):

I think it remains to be seen if the numbers above are transcendental. The Mathematica (version 10) command Element[z , Algebraics ] cannot determine whether these numbers belong to the domain of algebraic numbers or not.

Now let’s consider an expression containing π , e and i (the imaginary unit complex number). The following expressions are equivalent:

A 2D complex plot of the last function above (with z between -4 and 4) gives the following graph (made with Maple):

A 3 D complex plot of the same function ( with z between -4-4i and 4+4i) gives the following graph (with the help of Maple):

A general solution of the function f(z) above for f(z) = 0 is (calculated with Mathematica):

Here is an interesting result:

Using the Mathematica commands Element[z , Algebraics ] and Not[Element[z , Algebraics ]] , it seems that the solutions z of f(z)=0 above ( for different values of the constants ) do not belong to the domain of algebraic numbers , and are therefore transcendental numbers.

The (infamous , or famous , take your pick) sinc function is known to be one of two solutions of the differential equation:

This is a linear second order ordinary differential equation with dependent variable x and independent variable y.
I have tried to explore and find the solutions to this differential equation using mostly computer math software and programs.

The Texas Instruments 92 Plus scientific calculator and the Maple computer algebra system agree and give the same solution :

Mathematica gives the following solution:

After converting the exponentials to trigonometric functions the expression above and the solution of the differential equation given by Mathematica becomes:

This solution is less simple than (1). Note that if we make the assumption in Mathematica that λ > 0 ,we get the solution:

If we try to find the graphical representation of solution (1) for different values of the arbitrary constants c1 and c2 and with λ (or n ) between -3 and 3 ,we get the following graphs:

Then I tried to take the absolute value of λ in the numerator of (2) for different values of the arbitrary constants , and compared their graphs with the graphs of solution (1) for the same values of the constants. λ in (2) and n in (1) are between 1 and 4.

The graphical results are similar but not exactly the same.

Using the Manipulate built-in function in Mathematica , here is an animation of the graphs for solution (1) obtained by varying λ , c1 and c2:

Finally , I will give the solution of a generalized form of the differential equation above

obtained with Mathematica :

In the solution above , Jμ(x) is the Bessel function of the first kind , and Yμ(x) is the Bessel function of the second kind.

I’ll finish exploring the sinc function by showing a few 3D graphs of sinc related curves.
Here is a first set of 3D curves of sinc related functions:

This slideshow requires JavaScript.

As an example , here is the Mathematica code for the 3D curve √(x² + y²)sinc(√(x² + y²)) above:

And here is the second set of 3D surfaces related to sinc:

This slideshow requires JavaScript.

The Mathematica code for the curve given by sinc(x² -y²) is:

I’m somewhat fed up with sinc , so soon I will move on to other subjects.

Update: I will add a last group of six 3D curves related to sinc . They include curves where sinc as a function of x is multiplied by sinc as a function of y , such as sinc(x)×sinc(y) , sinc(ln(x))×sinc(ln(y)) with ‘ln’ the natural logarithm to the base e rendered as ‘log’ by Mathematica in the image , sinc(sin(x))×sinc(cos(y)) , and sinc(x²)×sinc(y²) .